Longest element of a Coxeter group

In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w0. See (Humphreys 1992, Section 1.8: Simple transitivity and the longest element, pp. 15–16) and (Davis 2007, Section 4.6, pp. 51–53).

Properties

A Coxeter group has a longest element if and only if it is finite; "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum.
The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order.
The longest element is an involution (has order 2: \( w_{0}^{{-1}}=w_{0}) \) , by uniqueness of maximal length (the inverse of an element has the same length as the element).[1]
For any w ∈ W , {\displaystyle w\in W,} w\in W, the length satisfies \( \ell (w_{0}w)=\ell (w_{0})-\ell (w)\).[1]
A reduced expression for the longest element is not in general unique.
In a reduced expression for the longest element, every simple reflection must occur at least once.[1]
If the Coxeter group is finite then the length of w0 is the number of the positive roots.[1]
The open cell Bw0B in the Bruhat decomposition of a semisimple algebraic group G is dense in Zariski topology; topologically, it is the top dimensional cell of the decomposition, and represents the fundamental class.
The longest element is the central element –1 except for \( A_{n}\) ( \( n\geq 2) \), \( D_{n} for n odd, \( E_{6}, \) and \( I_{2}(p) \) for p odd, when it is –1 multiplied by the order 2 automorphism of the Coxeter diagram. [2]